Indefinite Causal Order from Failure-to-Glue: Contextual Semantics and Parametric Time

Indefinite causal order (ICO) has been studied via higher-order quantum processes (e.g.\ the quantum switch), process matrices, and quantum-gravity proposals involving superposed causal structure, yet the meaning of "indefiniteness" and its relation to definite-order explanations often remain opaque. Part I develops a category-theoretic formulation of definite-order explainability as a gluing problem: each definite causal ordering (a partial order/DAG type) is treated as a context, and causal separability amounts to a consistent global section (possibly after convex mixing), whereas causal nonseparability is a failure-to-glue. We also introduce a compact seven-valued contextual classifier - an intuitionistic elaboration - that separates variation across contexts from genuine indeterminacy. Part II applies this framework to a quantum-gravity motivated setting where the fundamental time is a parametric ordering variable τ, distinct from geometric (spacetime) time. Adopting a stochastic-quantization perspective on spin-network dynamics (Hilbert space not assumed fundamental) and reading the Wheeler--DeWitt condition as an equilibrium/stationarity constraint, we interpret ICO as indeterminacy of the parametric order of coarse-grained relational interventions, even when the microscopic update process is globally ordered by τ. Together, the two parts provide a common language for comparing ICO criteria and for stating precisely what "no hidden definite order" means.